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Properties

Label 2-40-40.37-c2-0-8
Degree $2$
Conductor $40$
Sign $0.469 + 0.883i$
Analytic cond. $1.08992$
Root an. cond. $1.04399$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.88 − 0.665i)2-s + (−3.60 − 3.60i)3-s + (3.11 − 2.51i)4-s + (2.34 + 4.41i)5-s + (−9.20 − 4.40i)6-s + (1.47 + 1.47i)7-s + (4.20 − 6.80i)8-s + 17.0i·9-s + (7.36 + 6.76i)10-s + 11.3i·11-s + (−20.2 − 2.18i)12-s + (−3.17 − 3.17i)13-s + (3.77 + 1.80i)14-s + (7.46 − 24.3i)15-s + (3.39 − 15.6i)16-s + (−9.94 − 9.94i)17-s + ⋯
L(s)  = 1  + (0.943 − 0.332i)2-s + (−1.20 − 1.20i)3-s + (0.778 − 0.627i)4-s + (0.469 + 0.883i)5-s + (−1.53 − 0.733i)6-s + (0.211 + 0.211i)7-s + (0.525 − 0.850i)8-s + 1.89i·9-s + (0.736 + 0.676i)10-s + 1.02i·11-s + (−1.69 − 0.181i)12-s + (−0.244 − 0.244i)13-s + (0.269 + 0.128i)14-s + (0.497 − 1.62i)15-s + (0.212 − 0.977i)16-s + (−0.585 − 0.585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.469 + 0.883i$
Analytic conductor: \(1.08992\)
Root analytic conductor: \(1.04399\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1),\ 0.469 + 0.883i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.14742 - 0.689554i\)
\(L(\frac12)\) \(\approx\) \(1.14742 - 0.689554i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.88 + 0.665i)T \)
5 \( 1 + (-2.34 - 4.41i)T \)
good3 \( 1 + (3.60 + 3.60i)T + 9iT^{2} \)
7 \( 1 + (-1.47 - 1.47i)T + 49iT^{2} \)
11 \( 1 - 11.3iT - 121T^{2} \)
13 \( 1 + (3.17 + 3.17i)T + 169iT^{2} \)
17 \( 1 + (9.94 + 9.94i)T + 289iT^{2} \)
19 \( 1 + 11.2T + 361T^{2} \)
23 \( 1 + (-1.67 + 1.67i)T - 529iT^{2} \)
29 \( 1 - 41.1T + 841T^{2} \)
31 \( 1 + 29.2T + 961T^{2} \)
37 \( 1 + (8.60 - 8.60i)T - 1.36e3iT^{2} \)
41 \( 1 + 19.6T + 1.68e3T^{2} \)
43 \( 1 + (25.1 + 25.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-41.7 - 41.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-2.16 - 2.16i)T + 2.80e3iT^{2} \)
59 \( 1 - 38.9T + 3.48e3T^{2} \)
61 \( 1 + 87.5iT - 3.72e3T^{2} \)
67 \( 1 + (31.1 - 31.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 134.T + 5.04e3T^{2} \)
73 \( 1 + (26.1 - 26.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 23.1iT - 6.24e3T^{2} \)
83 \( 1 + (68.3 + 68.3i)T + 6.88e3iT^{2} \)
89 \( 1 + 75.2iT - 7.92e3T^{2} \)
97 \( 1 + (-57.5 - 57.5i)T + 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.51285154696498060887722527627, −14.28410135789219434609775539370, −13.15419580394178767301406451323, −12.23847160519881213707176851253, −11.27761907871928650070498923432, −10.23414015589555789944261513268, −7.22401538567307267776721834648, −6.44057022215485273945588674120, −5.10162403020970114977360192332, −2.14866135592246724033165561881, 4.15533150398560458259310121440, 5.24861001724935843194704241533, 6.30460981981086145719461711563, 8.685267142683938941750580452344, 10.40459626544187913733322395661, 11.40841887122812137354193056084, 12.53218431841721585694885306893, 13.83408843249506667607416378793, 15.22115789159155120129703700657, 16.25421744850531846839881824462

Graph of the $Z$-function along the critical line